Corollary

If $A$ is a finitely generated left $B$-module and $E$ a finitely generated left $A$-module, $\rho_*(E)$ is a finitely generated left $A$-module.

Let $C$ be a third ring, $\rho' : C \to B$ a ring homomorphism and $\rho'' = \rho \circ \rho'$ the composite homomorphism. It follows immediately from the definitions that $\rho''_*(E) = \rho'_*(\rho_*(E))$ for every $A$-module $E$. In particular, if $\rho$ is an isomorphism of $B$ onto $A$, then $E = \rho'_*(\rho_*(E))$, where $\rho'$ denotes the inverse isomorphism of $\rho$.